# Optimal penalty parameters for symmetric discontinuous Galerkin discretisations of the time-harmonic Maxwell equations

D. Sarmany, F. Izsak, Jacobus J.W. van der Vegt

Research output: Book/ReportReportProfessional

25 Citations (Scopus)

### Abstract

We provide optimal parameter estimates and a priori error bounds for symmetric discontinuous Galerkin (DG) discretisations of the second-order indefinite time-harmonic Maxwell equations. More specifically, we consider two variations of symmetric DG methods: the interior penalty DG (IP-DG) method and one that makes use of the local lifting operator in the flux formulation. As a novelty, our parameter estimates and error bounds are $i)$ valid in the pre-asymptotic regime; $ii)$ solely depend on the geometry and the polynomial order; and $iii)$ are free of unspecified constants. Such estimates are particularly important in three-dimensional (3D) simulations because in practice many 3D computations occur in the pre-asymptotic regime. Therefore, it is vital that our numerical experiments that accompany the theoretical results are also in 3D. They are carried out on tetrahedral meshes with high-order ($p = 1, 2, 3, 4$) hierarchic $H(\mathrm{curl})$-conforming polynomial basis functions.
Original language Undefined Enschede University of Twente, Department of Applied Mathematics 34 Published - Jan 2010

### Publication series

Name Memorandum / Department of Applied Mathematics Department of Applied Mathematics, University of Twente 1914 1874-4850 1874-4850

### Keywords

• METIS-270717
• Numerical mathematics
• EWI-17325
• Electromagnetic waves
• MSC-00A72
• Scientific computation
• Finite Element Method
• IR-69763

### Cite this

Sarmany, D., Izsak, F., & van der Vegt, J. J. W. (2010). Optimal penalty parameters for symmetric discontinuous Galerkin discretisations of the time-harmonic Maxwell equations. (Memorandum / Department of Applied Mathematics; No. 1914). Enschede: University of Twente, Department of Applied Mathematics.
Sarmany, D. ; Izsak, F. ; van der Vegt, Jacobus J.W. / Optimal penalty parameters for symmetric discontinuous Galerkin discretisations of the time-harmonic Maxwell equations. Enschede : University of Twente, Department of Applied Mathematics, 2010. 34 p. (Memorandum / Department of Applied Mathematics; 1914).
title = "Optimal penalty parameters for symmetric discontinuous Galerkin discretisations of the time-harmonic Maxwell equations",
abstract = "We provide optimal parameter estimates and a priori error bounds for symmetric discontinuous Galerkin (DG) discretisations of the second-order indefinite time-harmonic Maxwell equations. More specifically, we consider two variations of symmetric DG methods: the interior penalty DG (IP-DG) method and one that makes use of the local lifting operator in the flux formulation. As a novelty, our parameter estimates and error bounds are $i)$ valid in the pre-asymptotic regime; $ii)$ solely depend on the geometry and the polynomial order; and $iii)$ are free of unspecified constants. Such estimates are particularly important in three-dimensional (3D) simulations because in practice many 3D computations occur in the pre-asymptotic regime. Therefore, it is vital that our numerical experiments that accompany the theoretical results are also in 3D. They are carried out on tetrahedral meshes with high-order ($p = 1, 2, 3, 4$) hierarchic $H(\mathrm{curl})$-conforming polynomial basis functions.",
keywords = "METIS-270717, Numerical mathematics, EWI-17325, Electromagnetic waves, MSC-00A72, Scientific computation, Finite Element Method, IR-69763",
author = "D. Sarmany and F. Izsak and {van der Vegt}, {Jacobus J.W.}",
year = "2010",
month = "1",
language = "Undefined",
series = "Memorandum / Department of Applied Mathematics",
publisher = "University of Twente, Department of Applied Mathematics",
number = "1914",

}

Sarmany, D, Izsak, F & van der Vegt, JJW 2010, Optimal penalty parameters for symmetric discontinuous Galerkin discretisations of the time-harmonic Maxwell equations. Memorandum / Department of Applied Mathematics, no. 1914, University of Twente, Department of Applied Mathematics, Enschede.
Enschede : University of Twente, Department of Applied Mathematics, 2010. 34 p. (Memorandum / Department of Applied Mathematics; No. 1914).

Research output: Book/ReportReportProfessional

TY - BOOK

T1 - Optimal penalty parameters for symmetric discontinuous Galerkin discretisations of the time-harmonic Maxwell equations

AU - Sarmany, D.

AU - Izsak, F.

AU - van der Vegt, Jacobus J.W.

PY - 2010/1

Y1 - 2010/1

N2 - We provide optimal parameter estimates and a priori error bounds for symmetric discontinuous Galerkin (DG) discretisations of the second-order indefinite time-harmonic Maxwell equations. More specifically, we consider two variations of symmetric DG methods: the interior penalty DG (IP-DG) method and one that makes use of the local lifting operator in the flux formulation. As a novelty, our parameter estimates and error bounds are $i)$ valid in the pre-asymptotic regime; $ii)$ solely depend on the geometry and the polynomial order; and $iii)$ are free of unspecified constants. Such estimates are particularly important in three-dimensional (3D) simulations because in practice many 3D computations occur in the pre-asymptotic regime. Therefore, it is vital that our numerical experiments that accompany the theoretical results are also in 3D. They are carried out on tetrahedral meshes with high-order ($p = 1, 2, 3, 4$) hierarchic $H(\mathrm{curl})$-conforming polynomial basis functions.

AB - We provide optimal parameter estimates and a priori error bounds for symmetric discontinuous Galerkin (DG) discretisations of the second-order indefinite time-harmonic Maxwell equations. More specifically, we consider two variations of symmetric DG methods: the interior penalty DG (IP-DG) method and one that makes use of the local lifting operator in the flux formulation. As a novelty, our parameter estimates and error bounds are $i)$ valid in the pre-asymptotic regime; $ii)$ solely depend on the geometry and the polynomial order; and $iii)$ are free of unspecified constants. Such estimates are particularly important in three-dimensional (3D) simulations because in practice many 3D computations occur in the pre-asymptotic regime. Therefore, it is vital that our numerical experiments that accompany the theoretical results are also in 3D. They are carried out on tetrahedral meshes with high-order ($p = 1, 2, 3, 4$) hierarchic $H(\mathrm{curl})$-conforming polynomial basis functions.

KW - METIS-270717

KW - Numerical mathematics

KW - EWI-17325

KW - Electromagnetic waves

KW - MSC-00A72

KW - Scientific computation

KW - Finite Element Method

KW - IR-69763

M3 - Report

T3 - Memorandum / Department of Applied Mathematics

BT - Optimal penalty parameters for symmetric discontinuous Galerkin discretisations of the time-harmonic Maxwell equations

PB - University of Twente, Department of Applied Mathematics

CY - Enschede

ER -

Sarmany D, Izsak F, van der Vegt JJW. Optimal penalty parameters for symmetric discontinuous Galerkin discretisations of the time-harmonic Maxwell equations. Enschede: University of Twente, Department of Applied Mathematics, 2010. 34 p. (Memorandum / Department of Applied Mathematics; 1914).